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63
Vertex Cover Might be Hard to Approximate to within 2ɛ
 IN PROCEEDINGS OF THE 18TH ANNUAL IEEE CONFERENCE ON COMPUTATIONAL COMPLEXITY
, 2003
"... We show that vertex cover is hard to approximate within any constant factor better than 2 where the hardness is based on a conjecture regarding the power of unique 2prover1round games presented in [15]. We actually show a stronger result, namely, based on the same conjecture, vertex cover on k ..."
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Cited by 150 (11 self)
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We show that vertex cover is hard to approximate within any constant factor better than 2 where the hardness is based on a conjecture regarding the power of unique 2prover1round games presented in [15]. We actually show a stronger result, namely, based on the same conjecture, vertex cover on kuniform hypergraphs is hard to approximate within any constant factor better than k.
Linear vs. Semidefinite Extended Formulations: Exponential Separation and Strong Lower Bounds
, 2012
"... We solve a 20year old problem posed by Yannakakis and prove that there exists no polynomialsize linear program (LP) whose associated polytope projects to the traveling salesman polytope, even if the LP is not required to be symmetric. Moreover, we prove that this holds also for the cut polytope an ..."
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Cited by 54 (13 self)
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We solve a 20year old problem posed by Yannakakis and prove that there exists no polynomialsize linear program (LP) whose associated polytope projects to the traveling salesman polytope, even if the LP is not required to be symmetric. Moreover, we prove that this holds also for the cut polytope and the stable set polytope. These results were discovered through a new connection that we make between oneway quantum communication protocols and semidefinite programming reformulations of LPs.
Integrality gaps for strong SDP relaxations of unique games
"... Abstract — With the work of Khot and Vishnoi [18] as a starting point, we obtain integrality gaps for certain strong SDP relaxations of Unique Games. Specifically, we exhibit a Unique Games gap instance for the basic semidefinite program strengthened by all valid linear inequalities on the inner pro ..."
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Cited by 42 (8 self)
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Abstract — With the work of Khot and Vishnoi [18] as a starting point, we obtain integrality gaps for certain strong SDP relaxations of Unique Games. Specifically, we exhibit a Unique Games gap instance for the basic semidefinite program strengthened by all valid linear inequalities on the inner products of up to exp(Ω(log log n) 1/4) vectors. For a stronger relaxation obtained from the basic semidefinite program by R rounds of Sherali–Adams liftandproject, we prove a Unique Games integrality gap for R = Ω(log log n) 1/4. By composing these SDP gaps with UGChardness reductions, the above results imply corresponding integrality gaps for every problem for which a UGCbased hardness is known. Consequently, this work implies that including any valid constraints on up to exp(Ω(log log n) 1/4) vectors to natural semidefinite program, does not improve the approximation ratio for any problem in the following classes: constraint satisfaction problems, ordering constraint satisfaction problems and metric labeling problems over constantsize metrics. We obtain similar SDP integrality gaps for Balanced Separator, building on [11]. We also exhibit, for explicit constants γ, δ> 0, an npoint negativetype metric which requires distortion Ω(log log n) γ to embed into ℓ1, although all its subsets of size exp(Ω(log log n) δ) embed isometrically into ℓ1. Keywordssemidefinite programming, approximation algorithms, unique games conjecture, hardness of approximation, SDP hierarchies, Sherali–Adams hierarchy, integrality gap construction 1.
Integrality gaps for sheraliadams relaxations
 In Proceedings of the FortyFirst Annual ACM Symposium on Theory of Computing
, 2009
"... We prove strong lower bounds on integrality gaps of Sherali–Adams relaxations for MAX CUT, Vertex Cover, Sparsest Cut and other problems. Our constructions show gaps for Sherali–Adams relaxations that survive nδ rounds of lift and project. For MAX CUT and Vertex Cover, these show that even nδ rounds ..."
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Cited by 39 (2 self)
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We prove strong lower bounds on integrality gaps of Sherali–Adams relaxations for MAX CUT, Vertex Cover, Sparsest Cut and other problems. Our constructions show gaps for Sherali–Adams relaxations that survive nδ rounds of lift and project. For MAX CUT and Vertex Cover, these show that even nδ rounds of Sherali–Adams do not yield a better than 2 − ε approximation. The main combinatorial challenge in constructing these gap examples is the construction of a fractional solution that is far from an integer solution, but yet admits consistent distributions of local solutions for all small subsets of variables. Satisfying this consistency requirement is one of the major hurdles to constructing Sherali–Adams gap examples. We present a modular recipe for achieving this, building on previous work on metrics with a local–global structure. We develop a conceptually simple geometric approach to constructing Sherali–Adams gap examples via constructions of consistent local SDP solutions. This geometric approach is surprisingly versatile. We construct Sherali–Adams gap examples for Unique Games based on our construction for MAX CUT together with a parallel repetition like procedure. This in turn allows us to obtain Sherali–Adams gap examples for any problem that has a Unique Games based hardness result (with some additional conditions on the reduction from Unique Games). Using this, we construct 2 − ε gap examples for Maximum Acyclic Subgraph that rules out any family of linear constraints with support at most nδ. 1
Rounding Semidefinite Programming Hierarchies via Global Correlation
, 2011
"... We show a new way to round vector solutions of semidefinite programming (SDP) hierarchies into integral solutions, based on a connection between these hierarchies and the spectrum of the input graph. We demonstrate the utility of our method by providing a new SDPhierarchy based algorithm for constr ..."
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Cited by 36 (4 self)
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We show a new way to round vector solutions of semidefinite programming (SDP) hierarchies into integral solutions, based on a connection between these hierarchies and the spectrum of the input graph. We demonstrate the utility of our method by providing a new SDPhierarchy based algorithm for constraint satisfaction problems with 2variable constraints (2CSP’s). More concretely, we show for every 2CSP instance ℑ a rounding algorithm for r rounds of the Lasserre SDP hierarchy for ℑ that obtains an integral solution that is at most ε worse than the relaxation’s value (normalized to lie in [0, 1]), as long as r> k · rank�θ(ℑ) / poly(ε), where k is the alphabet size of ℑ, θ = poly(ε/k), and rank�θ(ℑ) denotes the number of eigenvalues larger than θ in the normalized adjacency matrix of the constraint graph of ℑ. In the case that ℑ is a Unique Games instance, the threshold θ is only a polynomial in ε, and is independent of the alphabet size. Also in this case, we can give a nontrivial
Lower bounds for LovászSchrijver systems and beyond follow from multiparty communication complexity
, 2006
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Linear Level Lasserre Lower Bounds for Certain kCSPs
"... We show that for k ≥ 3 even the Ω(n) level of the Lasserre hierarchy cannot disprove a random kCSP instance over any predicate type implied by kXOR constraints, for example kSAT or kXOR. (One constant is said to imply another if the latter is true whenever the former is. For example kXOR constr ..."
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Cited by 31 (1 self)
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We show that for k ≥ 3 even the Ω(n) level of the Lasserre hierarchy cannot disprove a random kCSP instance over any predicate type implied by kXOR constraints, for example kSAT or kXOR. (One constant is said to imply another if the latter is true whenever the former is. For example kXOR constraints imply kCNF constraints.) As a result the Ω(n) level Lasserre relaxation fails to approximate such CSPs better than the trivial, random algorithm. As corollaries, we obtain Ω(n) level integrality gaps for the Lasserre hierarchy of 76 − ε for VertexCover, 2 − ε for kUniformHypergraphVertexCover, and any constant for kUniformHypergraphIndependentSet. This is the first construction of a Lasserre integrality gap. Our construction is notable for its simplicity. It simplifies, strengthens, and helps to explain several previous results.
Improved approximation guarantees through higher levels of SDP hierarchies
 In Proceedings of the 11th International Workshop, APPROX
, 2008
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